curve #2953

y² + xy = x³ − 139850900315036148264999711022186105105x − 4805364953478674587663550097742533246761457869799982423

a-invariants: [1, 0, 0, -139850900315036148264999711022186105105, -4805364953478674587663550097742533246761457869799982423]
rank (lower bound): ≥ 8
torsion subgroup: ℤ/6ℤ
conductor (N): 26841118146961827757736036579027307510
discriminant (Δ): 175045529873088486037810052100954891106645675554441769591839684634800691243559797760516675582044489080371127779328000
Faltings height: 20.9944
naive height: 275.1145
primes of bad reduction: 2, 3, 5, 7, 11, 13, 17, 29, 53, 59, 61, 83, 101, 433, 563, 2339, 4643, 15107, 28349
regulator: 331114045.295381771050493409626337075323229645827348652
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:29:54 UTC
last updated: 2026-09-26 20:29:54 UTC

Witness: 8 independent points log in to add more points →

(20885062392873283082, -78639482173097025838962331045)
(641881605098000176814, -16259555594520457935446906247937)
(12261725728017976330661594/229441, -41871448477684738888946197333663722203/109902239)
(24078725825364590702, -102899213335006607875627535425)
(4306755891709038836201/64, 278233791598561023221555072818591/512)
(-495169472677836543536049814/44849809, 4177442617039521213407841635279874630659/300359170873)
(-111418105141036281080584/10609, 19101767500565451739638500876303411/1092727)
(-89786138354773293960688/271441, 910136268211453289839063865170312405/141420761)

Commentary

2-isogenous image of D-z6old678-033, rational kernel x=274887355699892655/4. Source: https://dujella.github.io/tors/z6old678.html. Independent witness points transported by the degree-2 isogeny. Witnesses saturated by exact rational halves of published and transported relations; quadratic-character independence is certified.

last edited by Natanael Wildner Fraga at 2026-09-26 20:29:54 UTC · history